TITLE: Finite Field Extensions and the Sum of the Elements in Proper Subextensions
QUESTION [5 upvotes]: Let $F$ be a finite field, and let $u,v$ be algebraic over $F$. Consider the fields $F(u,v),F(u)$ and $F(v)$. Must it be the case that $F(u,v) = F(u)$ or $F(u,v) = F(u+v)$?

REPLY [5 votes]: No: if $ \mathbb F_4=\mathbb F_2(a)$ and  $\mathbb F_8=\mathbb F_2(b)$ then
$$\mathbb F_2(a, b-a)=\mathbb F_{64} \neq F_2(a)=\mathbb F_4 \quad   \text {and}\quad  \mathbb F_2(a, b-a)=\mathbb F_{64}\neq F_2(a+(b-a))=\mathbb F_8$$